Journal of the Ramanujan Mathematical Society

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An Improved Derivative-Free Method Via Double Direction Approach for Solving Systems of Nonlinear Equations


Affiliations
1 epartment of Mathematics and Computer Science, Sule Lamido University, Kafin Hausa, Jigawa, Nigeria
2 Department of Mathematical Sciences, Bayero University Kano, Nigeria
     

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We present some double direction method for solving large-scale system of nonlinear equations. This method used the special form of iteration by introducing the two direction vectors in different ways. The approximation to the Jacobian matrix is done by sufficiently constructed diagonal matrix via acceleration parameter as well as derivative-free line search procedure. The proposed method is proved to be globally convergent under mild condition. Finally, numerical comparison using a large scale benchmark test problems show that the proposed method is very promising.
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  • N. I. Dbaruranovic-milicic and M. Gardasevic-Filipovic, A multi step curve search algorithm in nonlinear convex case, Facta Universitatis Series: Mathematics and Informatics, 25 (2010) 11–24.

  • N. I. Dbaruranovic-milicic, A multi step curve search algorithm in nonlinear optimization, Yugoslav Journal of Operation Research, 18 (2008) 47–52.

  • N. Andrei, An accelerated gradient descent algorithm with backtracking for unconstrained optimization, Numerical Algorithm, 42 (2006) 63–73.

  • J. E. Dennis, Jr. and R. B. Schnabel, Numerical method for unconstrained optimization and non-linear equations, Practice Hall, Englewood cliffs, NJ, USA (1983).

  • L. Zang, W. Zhou and D. H. Li, Global convergence of modified Fletcher-Reeves conjugate gradient method with Armijo-type line search, Numerische Mathematik, 164(1) (2005) 277–289.

  • L. Zang, W. Zhou and D. H. Li, A descent modified Polak-Pibiere-Polyak conjugate gradient method and its global convergence, IMA Journal of Numerical Analysis, 164(1) (2006) 227–572.

  • M. Y. Waziri and Jamilu Sabiu, A derivative-free conjugate gradient method and its global convergence for symmetric nonlinear equations, Journal of Mathematics and Mathematical Sciences, 8 (2015).

  • M. Y. Waziri, W. J. Leong, M. A. Hassan and M. Monsi, A new Newton’s method with diagonal Jacobian approximation for system of nonlinear equations, Journal of Mathematics and Statistics, 6(3), (2010) 246–252.

  • M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAMJournal of Optimization, 7 (1997) 26–33.

  • M. Raydan, On Barzilai and Borwein choice of step length for the gradient method, IMA Journal of Numerical Analysis, 13 (1993) 321–326.

  • D. Li and M. Fukushima, A global and superlinear convergent Gauss-Newton based BFGS method for symmetric nonlinear equation, SIAM Journal on Numerical Analysis, 37 (1999) 152–172.

  • M. J. Petrovic and P. S. Stanimirovic, Accelerated double direction method for solving unconstrained optimization problems, Mathematical Problem Eng. (2014).

  • Gongli Yuan∗ and Xiwen Lu, A new backtracking inexact BFGS method for symmetric nonlinear equations, Journal of Computers and Mathematics with Applications, 55 (2008) 116–129.

  • P. S. Stanimirovic and M. B.Miladinovic, Accelerated gradient descent methods with line search, Journal of Numerical Algorithm, 13 (1993) 321–326.

  • M. J. Petrovic, An accelerated double step size model in unconstrained optimization, Journal of Mathematics and Computation, 250 (2015) 309–319.

  • E. Dolan and J. More, Benchmarking opyimization software with performance profiles, Journal of Mathematical Program, 91 (2002) 201–213.

  • Qin-Rong Yana, Xiao-Zhen Penga and Dong-Hui Li, A globally convergent derivative-free method for solving large-scale nonlinear monotone equations, Journal of Computational and Applied Mathematics, 234 (2010) 649–657.

  • W. Zhou and D. Shen, An inexact PRP conjugate gradientmethod for symmetric nonlinear equations, Numerical Functional Analysis and Optimization, 35 (2014) 370–388.


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  • An Improved Derivative-Free Method Via Double Direction Approach for Solving Systems of Nonlinear Equations

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Authors

Abubakar S. Halilu
epartment of Mathematics and Computer Science, Sule Lamido University, Kafin Hausa, Jigawa, Nigeria
M. Y. Waziri
Department of Mathematical Sciences, Bayero University Kano, Nigeria

Abstract


We present some double direction method for solving large-scale system of nonlinear equations. This method used the special form of iteration by introducing the two direction vectors in different ways. The approximation to the Jacobian matrix is done by sufficiently constructed diagonal matrix via acceleration parameter as well as derivative-free line search procedure. The proposed method is proved to be globally convergent under mild condition. Finally, numerical comparison using a large scale benchmark test problems show that the proposed method is very promising.

References